J.J. Stoker - Water waves. The mathematical theory with applications, страница 102

See Hydraulic jumps.Forced oscillations, 55Junction of the Ohio and MississippiFourier integral theorem, 153Rivers, 457, 509Fourier transform, 35, 155flood wave through, 494Free natural vibrations of a lake, 424Free surface, 11. (See also Surface.)Kelvin's theory of ship waves, 219Free surface condition, 20Kinematic boundary condition, 16, 56SUBJECT INDEX565Lagrangian form of the equations of Open channel flows, 451motion, 4unsteady, 409Lake, free natural vibrations of, 424 Optics, geometrical, 133Levi-Civita's theory, 513, 522Oscillations, forced, 55Linear theory, 35free, 55derivation of, 19in harbors, 414free surface condition, 21of a lake, 423Local speed of small disturbance. Seeof the atmosphere, 374Wave speed.pitching, 250Long waves, theory of, 23, 291rolling, 250simple harmonic, 37Much lines, 40Usmall, 35Manning's formula, 454yawing, 250Manning's roughness coefficient, 457Overhanging cliff, 73Margules' law, 387Massflux,across a shock front, 318Mathematical hydraulics, 451Meteorology, 374hydrostatic pressure law in, 374, 382Michell's type ship, 257Microscisms, origin of, 137Mississippi Hivcr, 509flood waves, 458, 484model, 482, 509junction with the Ohio River, 457, 509Models of the Ohio and MississippiRivers, 482, 508Model studies of harbors, 429Momentum,conservationof,3Monoclinal flood wave, 4(52Motions.

(See also Flow; Wares.)steady, 199, 201uniqueness of unsteady, 187unsteady, 149Moving pressure point, 217Non-existence of depression shock, 323Nonlinear free surface condition, 11, 513Non linearity of breaking phenomena, 71Nonlinear shallow water theory equations, 24Numerical solutions for sloping beaches,73, 75Obstacles, flows over, 344waves due to, 35Occluded front, 381Oceanography, 133Ocean tides, 421Ohio River, 505flood waves, 458, 484junction with the Mississippi River,457modelof,482Particle derivative, 5Periodic impulse, waves due to, 17 1Periodic surface pressure, 57Periodic waves.

(See also Waves.)existence of, 522Perturbation procedure, 19, 269Phase speed, 170Pitching oscillation, 250, 278Point source (or sink), 12Potential, singularities of, 12Potential flow, 9Pressure, 3. (Sec also Surface pressure.)periodic, 57waves caused by moving, 219Privaloff's theorem, 536Profile, of a river, 458Progressing waves, 57. (See also Wares.)of fixed shape, 461simple harmonic, 45Propagation of discontinuities alongcharacteristics, 474Propagation speed. See Wave speed.Radiation condition, 174, 209. (See alsoSommerfeld entries. )Range of influence, 299Rnyleigh-Uitz method, 426Reflection of energy, 446of shock, 330of waves, 71, 95Refraction along a coast, 133Resistance force, 453Resonance, 58Rigid body, floating, 245River profile, 458Rivers, backwater effects in, 461Rolling oscillations, 250Roll waves, 466Roughness coefficient, 454, 466Manning, 457WATER WAVES566Running stream, waves ona,198in a dam, flow through,Small amplitude theory, 19.

(SeeLinear theory.)Small oscillations. See Oscillations.Solitary wave, 327, 342, 370approximation, 343experimental work, 351Sommerfeld's diffraction problem,Sommerfeld's radiation condition,65, 111, 113, 175SluiceSchauder-Leray theory, 522Schmidt, E., bifurcation theory of, 529Seiche, 423Seismology, 137Shallow water, floating bodies, 431long waves, 291Shallow water theory, 22, 291accuracy, 27equations of, 24for sloping beaches, 75higher-order approximation in, 28, 32linear, 25, 75, 414linear,compared with numericalsolution, 75linear sound speed, 419mathematical justification, 31reformulation of equations, 292systematic derivation of, 27validity beyond the breaking point,362Ship, as a floating rigid body, 245Ship wave problem, 219, 224Ship waves, diverging system, 237in water of finite depth, 243method of stationary phase, 219transverse system, 237Shock, 317advancing into still water, 323back of, 321conditions, 314constant, 326energy balance across, 318front of, 321mass flux across, 318non-existence of depression, 323reflected from a rigid wall, 330resulting from the breaking of a dam,333turbulence at front of, 320Simple harmonic waves, 37energy transmission, 47progressing, 45Simple wave, 300, 469applications to problems of meteorology, 391centered, 311Sound speed,26.

(See also12, 13numerical solutions, 73, 7510959,speed.)Source, 12Standingwaves,cylindricallysym-metric, 41simple harmonic, 37three-dimensional, 41two-dimensional, 38Stationary front, 378Stationary phase, 163, 219justification, 181Steady flow, supercritical, 405with critical speed, 344Steady motions, 199Steady state problems, unnaturalness,175Stoke's phenomenon, 117Stoke's theorem, 8Stream, waves on a running, 198Surge, 250Sway, 250Subcritical flow, 305, 406Supercritical flow, 304, 406supersonic flow, 304Surface, 11. (See also Free surface.)condition, exact, 11, 513disturbance, motions due to, 156pressure, confined to a segment, 58periodic, 57simple harmonic, 55Surfacesof discontinuityintheat-mosphere, 375Surface waves, 18typical problem, 15Tidal oscillationsof the atmosphere,375Tidal theory, 22boundary conditions, 422Tides in the oceans, 421Transverse system of ship waves, 237Trim of a ship, 251Turbulence, 453alsoat a shock, 320Singularities of the velocity potential,Sink, 12Slenderness parameter, 250Sloping beaches, 69, 369.

(SeeBeaches.)experiments on, 71, 73, 75, 373Wave407alsoUndertow, 71Uniqueness, 150, 187SUBJECT INDEXVelocity, group, 170of flow of energy, 49Velocity potential, 9singularities of, 12Verticalcliff,three-dimensional wavesagainst a, 84Vertical wedge. See Diffraction arounda vertical wedge.Viscosity, 3Vibrations, of a lake, 424WarmWaterWavefront,380table experiments, 412motions. (See also Flow; Motions;and Waves. )inonopen channels, 451567energy transmission, 47existence of periodic, 522experimental work on solitary, 351experiments on sloping beaches, 71,73, 75group, 51initial, steepness, 357in open channels, 451internal, 147in the atmosphere, 374of small amplitude, 19on sloping beaches, 69, 369past obstacles, 69progressing, 57, 67progressing, of fixed shape, 461progressing, simple harmonic, 45reflection from shore, 71, 95roll, 466ship.

See Ship waves.simple, 300, 469simple harmonic standing, 37discontinuity surfaces in theatmosphere, 375Wave refraction along a coast, 133Wave resistance integral, 284Waves. (See also Flew; Motions.)against a vertical cliff, 67, 84solitary, 327, 342, 870breaking of, 69, 307standing. See Standing waves.breaking of at crests, 369steady, 207transverse system, 237breaking of on shallow beaches, 352breaking point, 354unsteady, 210Wave speed, 26, 293, 299, 473centered simple, 311in linear shallow water theory, 419cnoidal, 342in meteorology, 404depression, 306, 352Wave tanks, experiments in, 71, 73diverging system, 237due to a moving pressure point, 219 Wetted perimeter, 454due to disturbances from rest, 35Wiener-Hopf technique, 108, 141due to harmonic surface pressure, 49 Wine glass effect, 74due to obstacles in a running stream,35Yawing oscillation, 250due to periodic impulse, 174.